Question

The number of straight lines that can be drawn out of $10$ points of which $7$ are collinear is?

• A. $22$
• B. $23$
• C. $24$
• D. $25$

Answer 1

The correct option is D

$25$

Explanation for the correct option:

We have been given that there are $10$ points of which $7$ are collinear.

We know that to make a line we need at least $2$ points. So, the number of ways to select $2$ points from $10$ points to make straight lines is ${}^{10}C_2$.

As it is given that $7$ points are collinear which means that we can’t make lines for all the $7$ points. So, the number of ways to select $2$ points from these $7$ points which will not give any distinct line is ${}^{7}C_2$

So, we will subtract$${}^7{C}_2$$ from ${}^{10}C_2$. But $7$ collinear points are on a straight line. Therefore we have to add $$+1$$ in ${}^{10}C_2$

$$\therefore$$Total number of lines $={}^{10}C_2-{}^{7}C_2+1$

$$\begin{gathered} =\frac{10!}{2!8!}-\frac{7!}{2!5!}+1\left[\because {}^{n}C_r=\frac{n!}{r!\left(n-r\right)!}\right] \\ =\frac{10\times 9\times 8!}{2\times 1\times 8!}-\frac{7\times 6\times 5!}{2\times 1\times 5!}+1 \\ =45-21+1 \\ =46-21 \\ =25 \end{gathered}$$

Thus there are $25$ straight lines that can be drawn out of $10$ points of which $7$ are collinear.

Hence, option D is the correct answer.